This is an empty permanent load stage too however at this point the analysis of creep should be done, it’s a load where I will be able to compare to stage 8 and see the effects creep has on the superstructure in 100 years, at this point SLS should be covered to be consider a functional and successful structure.
Results and analysis of tendon proposal
Out of the results obtained on the previous analysis, the results to be analyzed are presented next. If we analyze the previous results obtained on the iterations in the cable layout proposal we can see that the layout provided is probably not the best option, in figure 186 I introduce the final moment diagram obtained at stage 8, at this point we should cover the moment diagram of selfweight provided by the program.
FIGURE 186: MOMENT DIAGRAM STAGE 8 4G-19T
as we can see we have tensions still on some members of the superstructure. The basic principle of the prestress establishes that we will add external forces to remove the loading effects, which in this specific case the loading effect being analyze is the
selfweight. This is a first indicator that the amounts of tendons proposed are not enough.
Also, we need to focus on the fact that the strut element is absorbing the effects needed, this is due to the high stiffness the element has. However, I will try to increase the area of prestress to see how the structure will act.
FIGURE 187: REACTIONS
At figure 187 we can see the reactions, for the design of the bridge the bearings on section 0 and 60 need to be able to hold 48.30 MN, as expected the reactions at section 20 and 40 are bigger since it distributes all the loading of the superstructure, the footings should be design for these values.
Iteration results
I will use the same method previously used on the project to find a solution, I will change the amount of prestress by iterations to see if the design proposal is correct and can provide a suitable solution for the service stages.
First iteration
At this iteration I have incremented the area of prestress from 8 groups to a total of 20 groups, where each group will have 40 tendons instead of 19.
Moment diagrams analysis
FIGURE 188: MOMENT DIAGRAM DUE TO PRESTRESS 20G-40T
FIGURE 189: MOMENT DIAGRAM STAGE 8 20G-40T
The first two figures are introduced to show us the effects the amount of prestress has on the structure, figure 188 shows us that the prestress works and keeps the shape, however the values we have only by this are not enough to cover the selfweight moment diagram. This shows us this is not a solution for our bridge. However, is important to notice that the values on the strut have incremented showing us that the strut element still holds most of the effects needed on the superstructure. If we analyze why this effect is done we can see that the primary forces on figure 175 are mainly focus on the area, considering that the cable layout proposal is based on a 45-degree angle to obtain the curvature point of the tendons these effects are facing the strut element, since it’s not a bearing point but a fixed part of the structure, the effects provided are translated to the strut.
If we compare the values on the center span of the superstructure on section 30 at figure 189, we can see that they have decrease by 54%, however we can still see that we have tension on the inner span of the superstructure. This has confirmed that the tendon is not a suitable solution.
SLS Check
FIGURE 190: STAGE 8 STRESSES ALL FIBERS 20G-40T
FIGURE 191: STAGE 10 STRESSES ALL FIBERS 20G-40T
Figure 190 and 191 show us the member’s stresses throughout the whole structure in the two main service loads. In the SLS check the stresses should cover limit values established by the concrete strength at service life. Since we are using concrete C 60/75 our limiting conditions are:
1. 𝜎 ≤ 0
𝑎𝑡 𝑡𝑜𝑝 𝑜𝑟 𝑏𝑜𝑡𝑡𝑜𝑚 𝑓𝑖𝑏𝑒𝑟𝑠 𝑡𝑜 𝑒𝑛𝑠𝑢𝑟𝑒 𝑝𝑢𝑟𝑒 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑖𝑜𝑛
2. |𝜎| ≤ 0.6 𝑓𝑐𝑘 = 0.6 ∗ 60 = 36 𝑀𝑃𝑎
𝑎𝑡 𝑡𝑜𝑝 𝑎𝑛𝑑 𝑏𝑜𝑡𝑡𝑜𝑚 𝑓𝑖𝑏𝑒𝑟𝑠 𝑡𝑜 𝑒𝑛𝑠𝑢𝑟𝑒 𝑝𝑢𝑟𝑒 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑖𝑜𝑛
From what we can see on the figures these two conditions are meet, showing us that SLS is checked, our biggest value is 23.4 in pure compression.
However, we need to notice that these graphs show us an envelope of the biggest stresses the superstructure has at each lamella. Reason why a check should be done on each fiber, figures 192-195 shows us the results for each stage and each fiber.
FIGURE 192: STAGE 8 TOP FIBERS STRESSES 20G-40T
FIGURE 193: STAGE 10 TOP FIBERS STRESSES 20G-40T
If we analyze the top fibers we can see that we have a problem on the triangular frame, we have tension fibers on both stages, is important to show that the effect of creep is also present, we can see that at the end of service life the stresses on the top fibers in the tension area have increment from 3.6 to 6.6 at the most critical point, and in compression stresses from -15.1 to -10.7 in the most critical point. We can see that we are not covering the limits, so SLS check fails here two. I need to provide more prestress area at the triangular frame, or in it defect avoid the strut element to take most of the effects to redistribute them in the superstructure.
FIGURE 194: STAGE 8 BOTTOM FIBERS STRESSES 20G-40T
FIGURE 195: STAGE 10 BOTTOM FIBERS STRESSES 20G-40T
Analyzing the bottom stages, we can see that we have the same problem of tension stresses, however at this point they are present in the inner span. We can see the creep effects here too since the values at stage 8 are negative, meaning compression, however at the end of service life they increment to 4.1 in tension, showing the solution provided is not enough. In case we were allowed to have a limited tension, these values should be lower than the concrete tensile strength.
Analysis first iteration
As we have seen on the results the solution provided is not enough to work with. It’s important to notice that I have doubled the amount of prestress and yet I haven’t been able to cover the limits for SLS and the tension moments that are still present in the structure, this will cause cracking on the concrete, which is one of the main principles of prestress, eliminating this effect mention get us able to have smaller cross sections since the elements will be stiffer, also it will prevent external agents to attack the steel.
New continuity tendon layout
It’s important to take in consideration the fact that I have analyze the section with a bigger number of tendons, this difficult the location of them in the cross-section with the design I provide, considering the minimum cover and distance between them, the center of gravity of the location of the tendons and the actual location in reality will differ, not providing the values we expect that the program gives us. We need to remember that all
tendons have a minimal and a maximal limit where they provide the effect. Reason why my suggestion is to add not bounded tendons in the inner span on the bottom fibers, this way I will try to eliminate the tension on the area, which will also redistribute the stresses at the end of the service life on the bottom fibers. I will still use the continuity cable proposal I introduce previously, since the shape has shown its effects on the structure. The new cables will start from the center of gravity of the lamella where it will be introduced. The new tendons will be presented on figure 196.
FIGURE 191: TENDON PROPOSAL 2
Results
I have provided as a comparison some results that I have obtained with the addition of 2 groups of 19 tendons each. However, for this example I have decrease the number of groups and tendons in the bonded continuity cable to 20 groups of 19 tendons each. At
1571 653
this point I have corrected the prestress at both ends. The final results are shown on the next figures.
FIGURE 192: PRIMARY FORCES UNBOUNDED TENDONS
FIGURE 193: SECONDARY FORCES UNBOUNDED TENDON
The first thing we need to notice out of the primary and the secondary forces is that we can see that the losses of prestress have been corrected at the end of the beam, this is due to the fact that the prestress will be done from both end. As we see the shape still keeps the same way as the previous analysis, however, we can see that the values have increased by 3 times on the inner span, even though we have decrease the cables on the bonded continuity cable. This shows us that the effects we are looking for in the inner cable are working due to the new layout. On figure 193 we can see that the effects on the compression at section 20 have also increased to the double values, this is due to the fact that here we consider more prestress than at figure 176.
FIGURE 194: MOMENT DIAGRAM UNBOUNDED TENDONS
FIGURE 195: MOMENT DIAGRAM UNBOUNDED TENDONS STAGE 8
on the previous figures we can see that the values of tendon due to the prestress have increase, is important to notice that we are providing less prestress area, however we get better results. Still as we see on figure 195, tensions are present on the inner span beam, however they have been lowered, this has shown us that this is not a suitable solution since we haven’t got rid of the tensions.
FIGURE 196: STAGE 8 ALL STRESSES UNBONDED TENDONS
FIGURE 197: STAGE 10 ALL STRESSES UNBONDED TENDONS
The first thing we need to notice out of figures 196 and 197 is that the effects are shown in these figures due to the new tendon layout, the values have increased on the compression fibers. Also, we can see that the creep effects are the same; they change by 0.3 in the whole service stage. These effects are expected.
FIGURE 198: STAGE 8 TOP STRESSES UNBOUNDED TENDONS
FIGURE 199: STAGE 10 TOP STRESSES UNBOUNDED TENDONS
As we can see on the figures above introduced, at the end of service life on the 70-meter beams we have tension present, since we are trying to keep zero values on this we cannot accept these results. However, is important to notice they have keep the same values, this is due to the fact that the effects of the new tendons redistribute the forces in the bonded continuity tendon, reason why having less tendons in these cables have the same effect, also as previously stated we can see that the values are similar in both ends, this is a difference from the previous figures since the tension is applied from both ends.
FIGURE 200: STAGE 8 BOTTOM STRESSES UNBOUNDED TENDONS
FIGURE 201: STAGE 10 BOTTOM STRESSES UNBONDED TENDONDS
This last two figures show us the effects on the bottom fibers, we still have some tension present, however they have been lowered considerably, the effects of the new tendon layout is clear here.
Final Analysis
From the results shown on this new layout I can assume that this is a good solution to be following, however my suggestion can be adding a new tendon on the top fibers on the 70-meters beam. This will correct the tension on this part, also redistributing the forces of the bonded continuity tendon. Also, I consider this can be optimized to have lower tendons. As you can see on the annexes I haven’t include this second tendon layout since I haven’t been able to find the appropriate optimization. If this is reached with the new suggestions we can lower the number of tendons on the continuity cable, my suggestion will be 8 groups of 19 tendons each, this way the tendons will keep between the limits of the cross section and will still have positive effects.
As we saw, it’s possible to keep lowering the amount of cables in the cross section, however I believe is better to obtain a better cross section to have more benefits out of the continuity cables bonded or unbounded.
CONCLUSIONS
Also, is important to notice that the project haven’t brought a satisfying solution to the problem faced so is better to analyze the structure again, focusing on eliminating the stresses on the superstructure. I have found many mistakes made me be in the program due to the lack of experience at the time of using it, however the process is the main factor to be followed from this project.
It is highly recommended to start the project with a smaller and more symmetrical cross-section and a different lamella casting distance in the cantilever, in these designs I used a 7.5 meters lamella where it prove to be big and not recommendable, is suggested to use a 5 meters lamella. In addition, the strut element should be less stiff to avoid the effects it has on the continuity cable design, since it has been proved that it absorbs the effect desired on the superstructure. If we are able to implement a more symmetric cross section, we will be able to provide more compression area in the structure, this way I will reduce the stresses created internally since the leaver arm will be smaller, creating less internal moments that affect the final internal forces throughout the whole structure, if this is done less prestress will need to be provided too.
If its decided to continue the project in order to optimize it based on the results obtained, its recommended to change the geometry of the cross sections of the project,
the height taken in account were based on predesigns, the original bridge and the design solution presented on this project are completely different, I attempted to replicate the structural design using my own calculations for cross sections, tendons, construction stages and service loads. This project does not include the design of the abutment, the footings, passive steel at the cross section, the strut and tie elements; they were placed as a solution for the main problems faced on construction and service stages.
Another recommendation to take in account is that the center of gravity in the structural model applied on the program has a variety from what in reality is, as previously stated the center of gravity of the superstructure was placed on the top fibers of the cross section, when in reality they change on each section depending of the height of the haunch. Its suggested that the design of a structure as important as this, it should be done two different models, one where the center of gravity is in the top fibers and one where the real connection of the nodes are made, this way it can be compared the values until obtaining the real results. This was a 2D model with many limitations, my results have varied due to this problem, however, I consider this was the best approach taken since we try to replicate the real bridge.
Conclusions.
expand my knowledge on this field and know I consider myself more prepared for future problems I might face on the area.
I am well aware that the decisions made throughout the project might seem like not suitable solutions, however is important to notice that out of the many problems I faced, I was able to find solutions for them. This structure selected at the beginning of the project brought many complications due to the complexity of it, the first decisions made by me, where a mix of solutions out of the preliminary investigation I did, I thought I was having the best previous design to start off, however, it was completely different. At the end of this project I can see that the main problem faced was in the cross section of the structure, it was to rigid, it had such a stiffness in every element of it that the solutions where not the most efficient, as we can see on the results obtained and previously discussed.
This project required more time than the available time scale, is important to notice that a project of this size usually is calculated by a team of engineers and it takes more than 5 months to have a suitable solution for approval of the authorities, I believe that if I would have had more time I would be able to find better solutions for the bridge design.
The bridge project has been divided in two main parts, construction stages and service stages as previously stated. They are a compliment, where a solution has to be found for both of them I was able to find a solution for construction stages, based on iterations, investigations and application of theory learned on classes and in the process of this design, however I consider it can be improved if the recommendations made are taken in consideration, if we are able to have a better solution for the constructions stages it will reflect on better solutions for the service stage.
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